Cluster tori over \mathbbF₂, hexagonal moves on triangulations, and minimal coverings of cluster manifolds

2025/09/04 by Melesio, Daniel Pérez, Simental, José
#05C15 #13F60 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.04614

Abstract

We study cluster algebras over \mathbbF₂. By the Laurent phenomenon there is a map from the set of seeds of the cluster algebra to the corresponding cluster variety. We show that in type A, fibers of this map can be described in terms of certain edges of the universal polytope of triangulations of a polygon. Moreover, we show that there is a section of this map giving seeds whose corresponding cluster tori cover the cluster manifold over any field \mathbbF, but there are also sections giving seeds whose cluster tori do not cover the cluster manifold over any field \mathbbF \not≅ \mathbbF₂.

Citations

• Deep Points of Cluster Algebras
• Cluster deep loci and mirror symmetry
• Braid variety cluster structures, II: general type
• Braid variety cluster structures, I: 3D plabic graphs
• Cluster structures on braid varieties
• Cohomology of cluster varieties. II. Acyclic case
• Introduction to Cluster Algebras. Chapters 1-3
• Cluster algebras III: Upper bounds and double Bruhat cells
• The Laurent phenomenon

Related

• A and D type cluster algebras: Triangulated surfaces and friezes
• Birational geometry of cluster algebras
• Cluster algebras and Poisson geometry
• On cluster algebras arising from unpunctured surfaces
• On Structure of cluster algebras of geometric type, II: Green's equivalences and paunched surfaces